Both a mirror and a 90 degree prism are not reflecting light on to a screen (cycles)
You should use way more samples than 201 samples (which you used in your .blend file) to the green wall to illuminate properly in cycles.Thats because tre light source you are using in your scene is way too far and way too strong. Atleast when i put the lamp near the prism the green plate was brighter.
U can clearly see in my picture that with about 10K samples the green plate was very bright